Quasi-Geostrophic Single-Layer Models

Abstract

The crudest representation of currents and winds is that given by a single, constant-density fluid layer in relative motion with respect to the rotating Earth. In the case of the ocean, the layer is bounded from below by the sea floor and from above by the free surface of the sea. In the case of the atmosphere, the layer is bounded from below by the ground and from above by a hypothetical surface, above which the density of the atmosphere goes abruptly to zero. In both cases, the horizontal pressure gradient arises from the modulation of such surfaces with respect to the geoid, while the Coriolis force tends to arrange the flow in geostrophic balance with the pressure gradient.

The resulting motion is inertial, and potential vorticity is conserved. If Earth’s curvature is taken into account, for instance, by considering the beta-plane approximation, a fundamental consequence of potential vorticity conservation for motions crossing circles of latitude is the formation of Rossby waves. The above-described model is also the first step towards the formulation of the homogeneous model of wind-driven ocean circulation in which both the wind-stress forcing and vorticity dissipation are taken into account. In this case, the modulation of the surfaces which sandwich the geostrophic layer is ascribed, according to the classical Ekman theory, to the convergence/divergence of the marine current just below the free surface, caused by the wind stress, and just above the sea floor in the benthic layer, caused by friction. Also, the lateral diffusion of relative vorticity in the interior may be considered as a dissipative mechanism.

The solutions of the homogeneous model explain many fundamental phenomena of large-scale ocean circulation, in spite of the very simple picture which is usually adopted in describing the structure of the planetary wind field over the oceans. It is worth noting that the Ekman model of the benthic layer applies also, as it stands, to the lower atmosphere from the ground up to about one kilometre. The related convergence/divergence of the wind is responsible of the vertical motion of air masses, whose vicissitudes influence the weather.